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Θεώρημα Gelfond-Schneider
Θεώρημα των Gelfond-Schneider Gelfond-Schneider theorem, Theorems thumb|300px| [[Μαθηματικά Μαθηματικό Θεώρημα Μαθηματικά Θεωρήματα Νόμοι Φυσικής Εξισώσεις Μαθηματικό Αξίωμα Αριθμός Μαθηματικός Χώρος ]] - Θεώρημα των Μαθηματικών. Ετυμολογία Η ονομασία "Θεώρημα" σχετίζεται ετυμολογικά με το όνομα του μαθηματικού "[[]]". Περιγραφή In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. The Gelfond–Schneider theorem answers affirmatively Hilbert's seventh problem. Statement :If α and β are algebraic numbers with α ≠ 0,1 and if β is not a rational number, then any value of αβ = exp(β log α) is a transcendental number. Comments * The values of α and β are not restricted to real numbers; complex numbers are allowed (and never rational, even if both the real and imaginary parts are rational). * In general, αβ = exp(β log α) is multivalued, where "log" stands for the complex logarithm. This accounts for the phrase "any value of" in the theorem's statement. * An equivalent formulation of the theorem is the following: if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then (log γ)/(log α) is either rational or transcendental. This may be expressed as saying that if log α, log γ are linearly independent over the rationals, then they are linearly independent over the algebraic numbers. The generalisation of this statement to several logarithms of algebraic numbers is in the domain of transcendence theory. * If the restriction that α and β be algebraic is removed, the statement does not remain true in general. For example, :: {\left(\sqrt{2}^{\sqrt{2}}\right)}^{\sqrt{2}} = \sqrt{2}^{\sqrt{2} \cdot \sqrt{2}} = \sqrt{2}^2 = 2. :Here, α is √2√2 which (as proved by the theorem itself) is transcedental rather than algebraic. Similarly, if α = 3 and β = (log 2)/(log 3), which is transcendental, then αβ = 2 is algebraic. A characterization of the values for α and β which yield a transcendental αβ is not known. * Kurt Mahler proved the ''p''-adic analogue of the theorem: if α and β are in C''p'', the completion of the algebraic closure of Q''p'', and they are algebraic over Q, and if α − 1 p'' < 1}} and β − 1 ''p < 1}}, then (log''p''α)/(log''p''β) is either rational or transcendental, where log''p'' is the ''p''-adic logarithm function. Corollaries The transcendence of the following numbers follows immediately from the theorem: * The numbers 2^{\sqrt{2}} (the Gelfond–Schneider constant) and its square root \sqrt{2}^{\sqrt{2}}. * The number e^{\pi} = \left( e^{i \pi} \right)^{-i} = (-1)^{-i} = 23.14069263 \ldots (Gelfond's constant), as well as i^i = \left( e^{i \pi / 2} \right)^i = e^{-\pi / 2} = 0.207879576 \ldots. See also Υποσημειώσεις Εσωτερική Αρθρογραφία * Μαθηματικά * Άλγεβρα * Γεωμετρία * Μαθηματική Ανάλυση * Τοπολογία * Μαθηματικό Θεώρημα * Μαθηματικά Θεωρήματα * Μαθηματικό Αξίωμα * Μαθηματικός Χώρος * Θεώρημα Lindemann-Weierstrass * Θεώρημα Baker; an extension of the result * Εικασία Schanuel; if proven it would imply both the Gelfond–Schneider theorem and the Lindemann–Weierstrass theorem Βιβλιογραφία * * * * * * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *A proof of the Gelfond–Schneider theorem *[ ] Category: Θεωρήματα Μαθηματικών